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Quantum textures of the many-body wavefunctions in magic-angle graphene

Nature volume 620, pages 525–532 (2023)Cite this article

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Abstract

Interactions among electrons create novel many-body quantum phases of matter with wavefunctions that reflect electronic correlation effects, broken symmetries and collective excitations. Many quantum phases have been discovered in magic-angle twisted bilayer graphene (MATBG), including correlated insulating1, unconventional superconducting2,3,4,5 and magnetic topological6,7,8,9 phases. The lack of microscopic information10,11 of possible broken symmetries has hampered our understanding of these phases12,13,14,15,16,17. Here we use high-resolution scanning tunnelling microscopy to study the wavefunctions of the correlated phases in MATBG. The squares of the wavefunctions of gapped phases, including those of the correlated insulating, pseudogap and superconducting phases, show distinct broken-symmetry patterns with a √3 × √3 super-periodicity on the graphene atomic lattice that has a complex spatial dependence on the moiré scale. We introduce a symmetry-based analysis using a set of complex-valued local order parameters, which show intricate textures that distinguish the various correlated phases. We compare the observed quantum textures of the correlated insulators at fillings of ±2 electrons per moiré unit cell to those expected for proposed theoretical ground states. In typical MATBG devices, these textures closely match those of the proposed incommensurate Kekulé spiral order15, whereas in ultralow-strain samples, our data have local symmetries like those of a time-reversal symmetric intervalley coherent phase12. Moreover, the superconducting state of MATBG shows strong signatures of intervalley coherence, only distinguishable from those of the insulator with our phase-sensitive measurements.

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Fig. 1: Imaging atomic-scale Kekulé patterns in MATBG.
Fig. 2: Symmetry-based order parameter decomposition and inter-Chern-sector coherence.
Fig. 3: Distinguishing correlated insulators at v = ±2 via moiré translation symmetry and IVC isospin vortices.
Fig. 4: Candidate theoretical ground states.
Fig. 5: Doping correlated insulators into superconducting and pseudogap phases.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code that supports the findings of this study is available from the corresponding author upon reasonable request.

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Acknowledgements

We thank O. Vafek, X. Liu, C.-L. Chiu and G. Farahi for discussions; and H. Ding for technical discussion. This work was primarily supported by the Gordon and Betty Moore Foundation’s EPiQS initiative grants GBMF9469 and DOE-BES grant DE-FG02-07ER46419 to A.Y. Other support for the experimental work was provided by NSF-MRSEC through the Princeton Center for Complex Materials NSF-DMR- 2011750, NSF-DMR-1904442, ARO MURI (W911NF-21-2-0147) and ONR N00012-21-1-2592. T.S. was supported by a fellowship from the Masason Foundation, and by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. J.P.H. was supported by the Princeton University Department of Physics. M.P.Z. was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract no. DE-AC02-05CH11231, within the van der Waals Heterostructures Program (KCWF16), and the Alfred P. Sloan Foundation. D.C., B.A.B. and N.R. were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 101020833), the ONR grant no. N00014-20-1-2303, Simons Investigator grant no. 404513, the Gordon and Betty Moore Foundation through the EPiQS Initiative, grant no. GBMF11070 and grant no. GBMF8685, NSF-MRSEC grant no. DMR-2011750, BSF Israel US foundation grant no. 2018226, and the Princeton Global Network Funds. J.H.-A. was supported by a Hertz Fellowship. N.R. acknowledges support from the QuantERA II Programme that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 101017733. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant JPMXP0112101001, JSPS KAKENHI grants 19H05790 and JP20H00354.

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Author notes

  1. These authors contributed equally: Kevin P. Nuckolls, Ryan L. Lee, Myungchul Oh, Dillon Wong, Tomohiro Soejima

Authors and Affiliations

  1. Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ, USA

    Kevin P. Nuckolls, Ryan L. Lee, Myungchul Oh, Dillon Wong, Jung Pyo Hong, Dumitru Călugăru, Jonah Herzog-Arbeitman, B. Andrei Bernevig & Ali Yazdani

  2. Department of Physics, University of California, Berkeley, CA, USA

    Tomohiro Soejima & Michael P. Zaletel

  3. Donostia International Physics Center, Donostia-San Sebastian, Spain

    B. Andrei Bernevig

  4. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

    B. Andrei Bernevig

  5. Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  6. International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

  7. Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France

    Nicolas Regnault

  8. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Michael P. Zaletel

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Contributions

K.P.N., R.L.L., M.O., D.W. and A.Y. designed the experiment. D.W., K.P.N., M.O. and R.L.L. fabricated the devices used for the study. M.O., R.L.L., K.P.N. and D.W. carried out STM and STS measurements. T.S., J.P.H. and M.P.Z. designed the order parameter decomposition scheme and built the software suite capable of performing this decomposition on STM data, with input from all authors. R.L.L., D.W., K.P.N. and M.O. performed the data analysis using this software suite and maintained the code. T.S., J.P.H., D.C. and J.H.-A. performed simulations of the LDOS of candidate insulating ground states under the guidance of B.A.B., N.R. and M.P.Z. K.W. and T.T. synthesized the hBN crystals. All authors discussed the results and contributed to the writing of the paper.

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Correspondence to Ali Yazdani.

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Nature thanks Peter Nemes-Incze, Xiao Yan Xu and Long-Jing Yin for their contribution to the peer review of this work.

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Nuckolls, K.P., Lee, R.L., Oh, M. et al. Quantum textures of the many-body wavefunctions in magic-angle graphene. Nature 620, 525–532 (2023). https://doi.org/10.1038/s41586-023-06226-x

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